Invertible linear preservers of semipositive matrices - a dimension free approach

Abstract

An m-by-n real matrix A is said to be semipositive if there exists a vector x>0 such that Ax>0, where the inequalities are understood componentwise. Dorsey et al. conjectured that any invertible linear map that $L$ that leaves invariant the collection of all semipositive matrices is always in the standard form $A\mapsto XAY$ for some row positive matrix $X$ and inverse nonnegative matrix $Y$. This was settled in when $m\ge n$. Our aim in this paper is to settle the case when m<n of the above conjecture. In fact, our proof works for arbitrary positive integers m and n. The main ingredient is a classification of affine subspaces of the largest possible dimension contained in the set of semipositive matrices.